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Advances in Mathematics ••• (••••) •••–•••www.elsevier.com/locate/aim

The canonical sheaf of Du Bois singularities

Sándor J. Kovács a,∗,1, Karl Schwede b,2, Karen E. Smith b,3

a University of Washington, Department of Mathematics, Seattle, WA 98195, USAb Department of Mathematics, University of Michigan, Ann Arbor, MI 48109-1109, USA

Received 29 May 2009; accepted 25 January 2010

Communicated by Ravi Vakil

In memoriam Juha Heinonen

Abstract

We prove that a Cohen–Macaulay normal variety X has Du Bois singularities if and only if π∗ωX′(G) �ωX for a log resolution π : X′ → X, where G is the reduced exceptional divisor of π . Many basic theoremsabout Du Bois singularities become transparent using this characterization (including the fact that Cohen–Macaulay log canonical singularities are Du Bois). We also give a straightforward and self-contained proofthat (generalizations of) semi-log-canonical singularities are Du Bois, in the Cohen–Macaulay case. It alsofollows that the Kodaira vanishing theorem holds for semi-log-canonical varieties and that Cohen–Macaulaysemi-log-canonical singularities are cohomologically insignificant in the sense of Dolgachev.© 2010 Published by Elsevier Inc.

1. Introduction

Consider a complex algebraic variety X. If X is smooth and projective, its De Rham complexplays a fundamental role in understanding the geometry of X. When X is singular, an analog of

* Corresponding author.E-mail addresses: [email protected] (S.J. Kovács), [email protected] (K. Schwede),

[email protected] (K.E. Smith).1 The author was supported in part by NSF Grants DMS-0554697 and DMS-0856185, and the Craig McKibben and

Sarah Merner Endowed Professorship in Mathematics.2 The author was partially supported by RTG grant number 0502170 and by a National Science Foundation

Postdoctoral Research Fellowship.3 The author was partially supported by NSF Grant DMS-0500823.

Please cite this article in press as: S.J. Kovács et al., The canonical sheaf of Du Bois singularities, Adv. Math. (2010),doi:10.1016/j.aim.2010.01.020

0001-8708/$ – see front matter © 2010 Published by Elsevier Inc.doi:10.1016/j.aim.2010.01.020


2 S.J. Kovács et al. / Advances in Mathematics ••• (••••) •••–•••

the De Rham complex, the Deligne–Du Bois complex plays a similar role. Based on Deligne’stheory of mixed Hodge structures, Du Bois defined a filtered complex of OX-modules, denotedby Ω ·

X , that agrees with the algebraic De Rham complex in a neighborhood of each smooth pointand, like the De Rham complex on smooth varieties, its analytization provides a resolution of thesheaf of locally constant functions on X [4].

Du Bois observed that an important class of singularities are those for which Ω0X , the zeroth

graded piece of the filtered complex Ω ·X , takes a particularly simple form (see Discussion 2.2).

He pointed out that singularities satisfying this condition enjoy some of the nice Hodge-theoreticproperties of smooth varieties. Dubbed Du Bois singularities by Steenbrink, these singularitieshave been promoted by Kollár as a natural setting for vanishing theorems; see, for example, [17,Ch. 12]. Since the 1980’s, Steenbrink, Kollár, Ishii, Saito and many others have investigated therelationship between Du Bois (or DB) singularities and better known singularities in algebraicgeometry, such as rational singularities and log canonical singularities. Because of the difficultiesin defining and understanding the Deligne–Du Bois complex Ω0

X , many basic features of DBsingularities have been slow to reveal themselves or have remained obscure. The purpose of thispaper is to prove a simple characterization of DB singularities in the Cohen–Macaulay case,making many of their properties and their relationship to other singularities transparent.

Let π : X → X be a log resolution of a normal complex variety X, and denote by G thereduced exceptional divisor of π . By Lemma 3.14 there exists a natural inclusion π∗ωX(G) ↪→ωX . The main foundational result of this article is the following multiplier ideal-like criterion forDB singularities:

Theorem 1.1 (= Theorem 3.1). Suppose that X is normal and Cohen–Macaulay. Let π : X′ → X

be any log resolution, and denote the reduced exceptional divisor of π by G. Then X has DBsingularities if and only if π∗ωX′(G) � ωX .

Theorem 1.1 is analogous to the following well-known criterion for rational singularities dueto Kempf: if X is normal and Cohen–Macaulay, then X has rational singularities if and only ifthe natural inclusion π∗ωX ↪→ ωX is an isomorphism. In particular, Theorem 1.1 immediatelyimplies that rational singularities are Du Bois, a statement that had been conjectured by Steen-brink in [31] and later proved by Kollár [17] in the projective case and finally by Kovács [20],and also independently by Saito [23] in general. Another immediate corollary is that normalquasi-Gorenstein DB singularities are log canonical; see Section 3 for a complete discussion.

In addition, this criterion shows that CM DB singularities relate to rational singularities verymuch like log canonical singularities relate to (kawamata) log terminal singularities. This hasbeen a general belief all along, but we feel that the criterion in Theorem 1.1 supports this beliefmore than anything else previously known.

A long-standing conjecture of Kollár’s predicts that log canonical singularities are Du Bois.Using Theorem 1.1, it is easy to see that Kollár’s conjecture holds in the Cohen–Macaulay case:

Theorem 1.2 (= Theorem 3.16). Suppose that X is normal and Cohen Macaulay, and that � isan effective Q-divisor on X such that KX + � is Q-Cartier. If (X,�) is log canonical, then X

has Du Bois singularities.

In fact, we prove the stronger result that Cohen–Macaulay semi-log canonical singularitiesare Du Bois, see Theorem 4.16 (even more, we prove that a generalization of semi-log canon-ical singularities are Du Bois). This is based on a technical generalization of certain aspects of



S.J. Kovács et al. / Advances in Mathematics ••• (••••) •••–••• 3

Theorem 1.1 to the non-normal case, treated in Section 4. Many special cases of Kollár’s conjec-ture had been known, including the isolated singularity case [12–14], the Cohen–Macaulay casewhen the singular set is not too big see also [20], and the local complete intersection case [25].

Very recently, Kollár’s conjecture that log canonical singularities are Du Bois, has been veri-fied by Kollár and the first named author, using recent advances in the Minimal Model Program[18]. In particular, there is now an independent and more general result proving Theorem 1.2.However, there are several reasons why Theorem 1.1 is still interesting (besides being the firstgeneral result of this kind). Kollár and Kovács also prove that the condition of being Cohen–Macaulay is constant in DB families. This means that a stable smoothable variety is necessarilyCohen–Macaulay, and hence the above condition is applicable.

Furthermore, Theorems 4.11 and 4.16, apply to non-normal singularities. These are the bestresults currently known. Notice that one of the main applications of DB singularities is to modulitheory and that is an arena where non-normal singularities may not be easily dismissed. Alreadyfor degenerations of curves one must deal with non-normal singularities. In particular, Kollár’sconjecture is actually important in the non-normal case, that is, we want to know that semi-logcanonical singularities are Du Bois.

Another immediate corollary, again predicted by Kollár, is that the Kodaira Vanishing The-orem holds for generalizations of (semi-)log canonical varieties. Fujino recently gave anotherproof of a closely related theorem using techniques of Ambro; see [7, Corollary 5.11].

Corollary 1.3 (= Corollary 6.6). Kodaira vanishing holds for Cohen–Macaulay weakly semi-logcanonical varieties. In particular, let (X,�) be a projective Cohen–Macaulay weakly semi-logcanonical pair and L an ample line bundle on X. Then Hi(X,L −1) = 0 for i < dimX.

Of course, if X is not Cohen–Macaulay, Kodaira vanishing in the above form necessarily fails.But the Cohen–Macaulay condition is not sufficient for Kodaira vanishing. Examples show thatsome further restriction on the singularities is needed; see [1, Section 2]. In some sense, thisis the most general form of the classical Kodaira vanishing theorem (that is, Hi(X,L −1) = 0for L ample and i < dimX) that could be hoped for.

We are also able to obtain some nice Hodge-theoretic properties for semi-log canonical singu-larities. In particular, we are able to show that Cohen–Macaulay semi-log canonical singularitiesare cohomologically insignificant in the sense of Dolgachev [3]; see Theorem 5.1. This fact hasuseful applications in the construction of compact moduli spaces of stable surfaces and higherdimensional varieties.

2. Preliminaries

In this section we will define the notion of log canonical, as well as DB singularities and statethe forms of duality we will use. Throughout this paper, a scheme will always be assumed tobe separated and noetherian of essentially finite type over C. By a variety, we mean a reducedseparated noetherian pure-dimensional scheme of finite type over C. Note that a variety mayhave several irreducible components. All varieties and schemes will be assumed to be quasi-projective. The purpose of this assumption is to guarantee that these varieties are embedded insmooth schemes. Note that in the end this hypothesis is harmless because implications betweenvarious types of singularities are local questions, thus the varieties may assumed to be quasi-projective.




We will use the following notation: For a functor Φ , RΦ denotes its derived functor on the(appropriate) derived category and RiΦ := hi ◦ RΦ where hi(C·) is the cohomology of thecomplex C· at the ith term. Similarly, Hi

Z := hi ◦ RΓZ where ΓZ is the functor of cohomologywith supports along a subscheme Z. Finally, Hom stands for the sheaf-Hom functor.

Let α : Y → Z be a birational morphism and � ⊆ Z a Q-divisor. Then α−1∗ � will denote theproper transform of � on Y .

2.1. Log canonical singularities

Let X be a normal irreducible variety of pure dimension d . The canonical sheaf ωX of X is theunique reflexive OX-module agreeing with the sheaf of regular differential d-forms

∧dΩX/C

on the smooth locus of X. A canonical divisor is any member KX of the (Weil) divisor classcorresponding to ωX . See Section 2.3 for the definition of the canonical sheaf on non-normalvarieties.

A (Q-)Weil divisor D is said to be Q-Cartier if, for some non-zero integer r , the Z-divisor rD

is Cartier, meaning that it is given locally as the divisor of some rational function on X. For sucha divisor, we can define the pullback π∗D, under any dominant morphism π , to be the Q-divisor1rπ∗(rD). We say that X is Q-Gorenstein if KX is Q-Cartier.

Now consider a pair (X,�), where � is an effective Q-divisor such that KX +� is Q-Cartier.In this case, there exists a log resolution of the pair; that is, a proper birational morphism from asmooth variety

π : X → X

such that Ex(π) is a divisor and furthermore the set π−1(�)∪Ex(π) is a divisor with simple nor-mal crossing support. Here Ex(π) denotes the exceptional set of π . Let � denote the birational(or proper, or strict) transform of � on X, often denoted by π−1∗ �. Then there is a numericalequivalence of divisors

KX + � − π∗(KX + �) ≡∑

aiEi

where the Ei are the exceptional divisors of π and the ai are some uniquely determined rationalnumbers. We can now define:

Definition 2.1. The pair (X,�) is called log canonical if ai � −1 for all i.

Definition 2.1 is independent of the choice of log resolution. For this and other details aboutlog resolutions, log pairs, and log canonical singularities see, for example, [19].

2.2. Du Bois singularities

As mentioned in the introduction, DB singularities are defined using a fairly complicatedfiltered complex Ω ·

X , which plays the role of the De Rham complex for singular varieties. Itfollows from the construction that there is a natural map (in the derived category of OX-modules)

OX → Ω0 ,


X



where Ω0X denotes the zeroth graded complex of the filtered complex Ω ·

X . By definition, X hasDu Bois (or DB) singularities if this map is a quasi-isomorphism. For a careful development ofthis point of view see [4,9] or [32]. However, a recent result of the second author in [25] pro-vides an alternate definition of Du Bois singularities, which we now review here (also comparewith [6]).

First, since the transverse union of two smooth varieties of different dimensions is an im-portant example of a DB singularity, we must leave the world of irreducible (and even that ofequidimensional) varieties, and instead consider reduced schemes of finite type over C. Recallthat a reduced subscheme X of a smooth ambient variety Y is said to have normal crossings if atevery point of x there exists a regular system of parameters for OY,x such that X is defined bysome monomials in these parameters.

Now suppose that X is a reduced closed subscheme of a smooth ambient Y . Let π : Y → Y

be a proper birational morphism such that:

(a) π is an isomorphism outside X,(b) Y is smooth,(c) π−1(X) has normal crossings (even though it may not be equidimensional).

Such morphisms always exist by Hironaka’s theorem; for example, π could be an embeddedresolution of singularities for X ⊂ Y , or a log resolution of the pair (Y,X). The conditions (a)–(c)can be relaxed somewhat; see [27, Proposition 2.20] for a way to relax condition (a) and [25] forfurther discussion. We set X to be the reduced preimage of X in Y , and again emphasize that X

may not be equidimensional. Under these conditions, Schwede shows that the object Rπ∗OX isnaturally quasi-isomorphic to the object Ω0

X defined in [4]. This leads to the following definition,equivalent to Steenbrink’s original definition:

Definition 2.2. We say that X has Du Bois (or DB) singularities if the natural map OX →Rπ∗OX is a quasi-isomorphism.

The fact that Definition 2.2 is independent of the choice of embedding and of the choiceof resolution simply follows from the fact that Rπ∗OX is quasi-isomorphic to the well-definedobject Ω0

X . See [25] for details (and again, compare with [6]).

2.3. Dualizing complexes and duality

For the convenience of the reader, we briefly review the form of duality we will use.Associated to every quasi-projective scheme X of dimension d and of finite type over C there

exists a (normalized) dualizing complex ω·X ∈ Db

coh(X), unique up to quasi-isomorphism. Toconstruct ω·

X concretely, one may proceed as follows. For a smooth irreducible variety Y ofdimension n, take ω·

Y = ωY [n], the complex that has the canonical module detΩY/C in degree−n and the zero module in all the other spots. Now, whenever X ⊆ Y is a closed embedding ofschemes of finite type over a field, we have

ω· = R Hom· (OX,ω· )

.


X Y Y



Because every quasi-projective scheme of finite type over C embeds in a smooth variety, thisdetermines the dualizing complex on any such finite type scheme. Alternatively, one can alsodefine ω·

X as h!C where h : X → C is the structural morphism. See [10] and [2] for details.For an equidimensional X admitting a dualizing complex one may define the canonical sheaf

to be the OX-module h−dimX(ω·X), denoted by ωX . If X is Cohen–Macaulay, the dualizing

complex turns out to be exact at all other spots (just as in the case of smooth varieties above). Inparticular, for a Cohen–Macaulay variety of pure dimension d , we have ω·

X = ωX[d]. If X is nor-mal and irreducible, h−dimX(ω·

X) agrees with the canonical sheaf ωX as defined in Section 2.1,so there is no ambiguity of terminology. More generally, if X is Gorenstein in codimension oneand satisfies Serre’s S2 condition, the canonical module h−dimX(ω·

X) = ωX is a rank one reflex-ive sheaf, and so corresponds to a Weil divisor class.

A very important tool we need is Grothendieck duality:

Theorem 2.3. (See [10, III.11.1, VII.3.4].) Let f : X → Y be a proper morphism between finitedimensional noetherian schemes. Suppose that both X and Y admit dualizing complexes and thatF · ∈ D−

qcoh(X). Then the duality morphism

Rf∗R Hom·X

(F ·,ω·

X

) → R Hom·Y

(Rf∗F ·,ω·

Y

)is a quasi-isomorphism.

Remark 2.4. In the previous theorem, ω·X should be thought of as f !ω·

Y .

3. Proof of the main theorem

In this section we prove our main result, a simple new characterization of DB singularities inthe normal Cohen–Macaulay case.

Theorem 3.1. Suppose that X is a normal Cohen–Macaulay variety. Let : X′ → X be any logresolution and denote the reduced exceptional divisor of by G. Then X has DB singularities ifand only if ∗ωX′(G) � ωX .

The proof of this theorem will take most of the present section. We will first show that it istrue for a special choice of log resolutions in Corollary 3.10. Then, in Lemma 3.12, we completethe proof by showing that the statement is independent of the choice of the log resolution.

The following notation will be fixed for the rest of this section:

Setup 3.2. Let X be a reduced equidimensional scheme of finite type over C embedded in asmooth variety Y , and let Σ denote a closed subscheme Σ � X that contains the singular locusof X. Assume that no irreducible component of X is a hypersurface, i.e., the codimension ofevery irreducible component of X in Y is at least two. Fix an embedded resolution π : Y → Y ofX in Y , and let X denote the strict transform of X on Y . Further assume that

(i) π is an isomorphism over X \ Σ , and(ii) π−1(Σ) is a simple normal crossing divisor of Y that intersects X in a simple normal cross-

ing divisor of X.




Let E denote the reduced preimage of Σ in Y , and X the reduced pre-image of X in Y . Weemphasize that X is not equidimensional; in fact, X is the transverse union of the smooth varietyX and the normal crossing divisor E. We will frequently abuse notation and use π to denote π |X .Note that π |X is a projective (respectively proper) morphism as long as π is. Finally, recall thatby [25] Rπ∗OX �qis Ω0

X and Rπ∗OE �qis Ω0Σ .

The outline of the proof of Theorem 3.1 goes as follows. Using the Grothendieck dual formof Schwede’s characterization of DB singularities stated in Lemma 3.3 below, it is clear that areduced Cohen–Macaulay scheme X of dimension d has DB singularities if and only if

(i) Riπ∗ω·X

= 0 for i �= −d , and

(ii) R−dπ∗ω·X

= ωX .

Our main technical statement is Theorem 3.8, which implies that for any normal variety of di-mension d , the sheaf R−dπ∗ω·

Xcan be identified with π∗ωX(G). This is proven by comparing

the dualizing complexes for X, X and E via the dual of the short exact sequence

0 → OX(−E|X) → OX → OE → 0.

On the other hand, the vanishing statement of (i) follows from a reinterpretation of a result of thefirst author [20] by the second author [26].

We first state the following dual form of Schwede’s characterization of DB singularities.

Lemma 3.3. X has DB singularities if and only if the natural map

Rπ∗ω·X

→ ω·X

is a quasi-isomorphism.

Proof. The result follows from Definition 2.2 via a standard application of Grothendieck dual-ity. �

Before beginning the proof of Theorem 3.1, we would like to make the following suggestiveobservation. If X has pure dimension d and Y has dimension n, then

h−d(ω·

X

) ∼= ωX(E|X),

h−n+1(ω·X

) ∼= ωE, and

hi(ω·

X

) = 0 for i not equal to − n + 1 or − d.

To see this, note that X and E have normal crossings and that there exists a short exactsequence,

0 → O˜(−E|˜) → O → OE → 0.


X X X



Next, dualize this sequence by applying R Hom·Y(_,ω

·Y) (and Grothendieck duality) to get an

exact triangle,

ω·E

ω·X

ω·X

⊗ OY (E)+1

.

Note that X is not equidimensional, but X and E are (in fact, they are Gorenstein and connected).Since E has dimension n − 1 and X has dimension d , we see that h−n+1(ω·

X) ∼= h−n+1(ω·

E),

proving the second statement. Taking the −d th cohomology proves the first statement since X

and E have normal crossings. It is easy to see that the third statement is true as well by takingany other cohomology. These three facts will not be used directly, but they do suggest a way toanalyze Rπ∗ω·

X.

With this in mind, we now prove that we really only need to understand the −d th cohomologyof Rπ∗ω·

X, at least in the Cohen–Macaulay case.

Proposition 3.4. In addition to (3.2) assume further that X is Cohen–Macaulay. Then X has DBsingularities if and only if the natural map

R−dπ∗ω·X

→ ωX

is surjective (if and only if it is an isomorphism).

Proof. If X has DB singularities, the statement (including the one in parentheses) follows triv-ially from Lemma 3.3.

Now assume that the natural map R−dπ∗ω·X

→ ωX is surjective, but X is not Du Bois. LetΣDB denote the non-Du Bois locus of X (cf. [20, 2.1]) and let x ∈ ΣDB a general point of(a component of) ΣDB. By [26, 5.11], the natural map

(Riπ∗ω·

X

)x

→ hi(ω·

X

)x

is injective for every i. The right side of this equation is zero for i �= −d since X is Cohen–Macaulay, and thus the left side is zero as well. For i = −d , the map is surjective by assumptionand, as we already noted, it is injective; hence it is an isomorphism. In particular, the localizedmap (Rπ∗ω·

X)x → (ω·

X)x is a quasi-isomorphism, contradicting Lemma 3.3 and the fact that(X,x) is not Du Bois. �Remark 3.5. Alternatively, one could use general hyperplane sections to reduce to the case of anisolated non-Du Bois point, and then apply local duality along with the key surjectivity of [21].

The following lemma will be important in the proof.

Lemma 3.6. Let Z be a reduced closed subscheme of Y , a variety of finite type over C. Thenhi(R Hom·

Y (Ω0Z,ω·

Y )) = 0 for i < −dimZ.

Proof. Without loss of generality, we may assume that Z and Y are affine. Let z ∈ Z be an ar-bitrary closed point. By local duality (see [10, V, Theorem 6.2] or [22, 2.4]), it is sufficient to




show that Hjz (Y,Ω0

Z) = Hjz (Z,Ω0

Z) = 0 for j > dimZ. We consider the hypercohomology spec-tral sequence H

pz (Z,hq(Ω0

Z)) that computes this cohomology. Note that dim(Supp(hq(Ω0Z))) �

dimZ−q by [9, V, 3.6], so that Hpz (Z,hq(Ω0

Z)) = 0 for p > dimZ−q (i.e., for p+q > dimZ).

Therefore, we see that Hjz (Z,Ω0

Z) vanishes for j > dimZ because every term in the spectral se-

quence that might possibly contribute to Hjz (Z,Ω0

Z) is zero. �Corollary 3.7. Riπ∗ω·

E = 0 for i < −dimΣ .

Proof. By [25] (cf. (3.2)) Rπ∗OE �qis Ω0Σ , so by Grothendieck duality

Rπ∗ω·E �qis R Hom·

Y

(Ω0

Σ,ω·Y

).

Then the statement follows from Lemma 3.6. �Theorem 3.8. If X is equidimensional and codimX Σ � 2, then R−dπ∗ω·

X∼= π∗ωX(E|X).

Remark 3.9. The codimension condition of this theorem implies that X must be R1. It is satisfied,for instance, if X is normal and the maximal dimensional components of Σ and SingX coincide.

Proof. Applying Rπ∗ to the exact triangle,

ω·E → ω·

X→ ω·

X⊗ OY (E) →

and taking cohomology leads to the exact sequence

R−dπ∗ω·E → R−dπ∗ω·

X→ R−dπ∗

(ω·

X⊗ OY (E)

) → R−d+1π∗ω·E.

The outside terms are zero by Corollary 3.7 which implies that the middle two terms are isomor-phic. To complete the proof, simply observe that

R−dπ∗(ω·

X⊗ OY (E)

) = R−dπ∗(ωX[d] ⊗ OY (E)

) = π∗(ωX ⊗ OY (E)

) = π∗ωX(E|X). �Corollary 3.10. In addition to (3.2) assume further that X is normal and Cohen–Macaulay. ThenX has DB singularities if and only if the natural map

π∗ωX(E|X) → ωX

(coming from (3.4) and (3.8)) is surjective (if and only if it is an isomorphism).

Remark 3.11. Note that π∗ωX(E|X) → ωX is an isomorphism on the smooth locus of X byconstruction, and hence it is always injective. For the same reason, this natural map is the sameas the one coming from Lemma 3.14.

At this point, we have proven Theorem 3.1 for a log resolution as in (3.2). The general casefollows from the next lemma which is well known to experts (for example, it also follows from[15, Lemma 1.6]).




Lemma 3.12. Let π : X′ → X be a proper birational morphism, Σ ⊆ X a closed subset anddenote by G the reduced pre-image of Σ via π . Assume that π is chosen such that X′ is smoothand G has simple normal crossings. Then π∗ωX′(G) on X is independent of the choice of π upto natural isomorphism.

Remark 3.13. This result is analogous to the fact that a multiplier ideal is independent of theresolution used to compute it.

Proof. Since any two proper birational morphisms mapping to X with the required propertiescan be dominated by a third such, it is sufficient to prove the following: Let X′′ be a smoothvariety and φ : X′′ → X′ a proper birational morphism and let H be the reduced pre-image of Σ

via π ◦ φ. Then φ∗ωX′′(H) � ωX′(G).The fact that X′ is smooth and G is a simple normal crossing divisor implies that the pair

(X′,G) has log canonical singularities. Furthermore, the support of the strict transform of G

on X′′ is contained in H by definition, and the rest of the components of H are φ-exceptional.Therefore,

ωX′′(H) � φ∗(ωX′(G))(F )

for an appropriate effective φ-exceptional divisor F . Applying the projection formula yields

φ∗ωX′′(H) � ωX′(G) ⊗ φ∗OX′′(F ),

and since F is effective and φ-exceptional, φ∗OX′′(F ) � OX′ [16, Lemma 1-3-2]. �We now turn our attention to using this criterion to prove that log canonical singularities are

Du Bois. First note that the statement is reasonably straightforward if X is Gorenstein so it wouldbe tempting to try to take a canonical cover, at least in the Q-Gorenstein case. However, it isnot clear that the canonical cover of a Cohen–Macaulay log canonical singularity is also Cohen–Macaulay. Examples of rational singularities with non-Cohen–Macaulay canonical covers in [28]suggest that this might be too much to hope for. Therefore, a different technique will be used.

Lemma 3.14. Let X be a normal irreducible variety and let : X′ → X be a log resolution of X.Let B be an effective integral divisor on X, B ′ = −1∗ B (the strict transform of B on X′), anddenote the reduced exceptional divisor of by G. Then there exists a natural injection,

∗ωX′(B ′ + G

)↪→ ωX(B).

Proof. Let ι : U ↪→ X denote the embedding of the open set over which is an isomorphism.As X is normal, we have that codimX(X \ U) � 2, and hence the following natural morphismsof sheaves:

∗ωX′(B ′ + G

)↪→ (

∗ωX′(B ′ + G

))∗∗ � ι∗(∗ωX′

(B ′ + G

)∣∣U

) � ωX(B),

where ( )∗ = HomX(_,OX) denotes the dual of a sheaf. The isomorphisms follow because X isS2 and is an isomorphism over U . �



Lemma 3.15. Let X be a normal irreducible variety with an effective Q-divisor D such that(X,D) has log canonical singularities, and let : X′ → X be a log resolution of (X,D). Let B

be an effective integral divisor on X with B � D. Denote the reduced exceptional divisor of byG and let B ′ = −1∗ B and D′ = −1∗ D. Then the following natural isomorphism holds:

∗ωX′(B ′ + G

) � ωX(B).

Proof. By Lemma 3.14 there exists a natural inclusion ι : ∗ωX′(B ′ + G) ↪→ ωX(B), so thequestion is local. We may assume that X is affine and need only prove that every section ofωX(B) is already contained in ∗ωX′(B ′ + G). Note that ι restricted to the naturally embeddedsubsheaf ωX′ ⊆ ωX′(B ′ + G) gives the usual natural inclusion ∗ωX′ ↪→ ωX .

Next, choose a canonical divisor KX′ and let KX = ∗KX′ . This is the divisor correspondingto the image of the section of ωX′ corresponding to KX′ via ι. As D′ = −1∗ D, it follows that thedivisors KX′ +D′ and −1∗ (KX +D) may only differ in exceptional components. We emphasizethat these are actual divisors, not just equivalence classes (and so are B and B ′).

Since X and X′ are birationally equivalent, their function fields are isomorphic. Let us identifyK(X) and K(X′) via ∗ and denote them by K . Further let K and K ′ denote the K-constantsheaves on X and X′ respectively.

Now we have the following inclusions:

Γ(X,∗ωX′

(B ′ + G

)) ⊆ Γ(X,ωX(B)

) ⊆ Γ (X,K ) = K,

and we need to prove that the first inclusion is actually an equality. Let g ∈ Γ (X,ωX(B)). Byassumption, B � D, so

0 � divX(g) + KX + B � divX(g) + KX + D. (3.15.1)

As (X,D) is log canonical and thus KX + D is Q-Cartier, there exists an m ∈ N such thatmKX + mD is a Cartier divisor and hence can be pulled back to a Cartier divisor on X′. Bythe choices we made earlier, we have that ∗(mKX + mD) = mKX′ + mD′ + Θ where Θ isan exceptional divisor. Again we emphasize that these are actual divisors and the two sides areactually equal, not just equivalent. We would also like to point out that Θ is not necessarilydivisible by m, neither is ∗(mKX + mD).

However, using the fact that (X,D) is log canonical, one obtains that Θ � mG. Combiningthis with (3.15.1) gives that

0 � divX′(gm

) + ∗(mKX + mD) � m(divX′(g) + KX′ + D′ + G

),

and in particular we obtain that

divX′(g) + KX′ + D′ + G � 0.

Claim. divX′(g) + KX′ + B ′ + G � 0.




Proof. By construction

divX′(g) + KX′ + B ′ + G = −1∗(

divX(g) + KX + B︸︷︷︸�0

) + F + G︸︷︷︸exceptional

. (3.15.2)

Where F is an appropriate exceptional divisor, though it is not necessarily effective. We alsohave that

divX′(g) + KX′ + B ′ + G = divX′(g) + KX′ + D′ + G︸︷︷︸�0

− (D′ − B ′)︸︷︷︸

non-exceptional

. (3.15.3)

Now let A be an arbitrary irreducible component of divX′(g) + KX′ + B ′ + G. If A were noteffective, it would have to be exceptional by (3.15.2) and non-exceptional by (3.15.3). Hence A

must be effective and the claim is proven. �It follows that g ∈ Γ (X′,ωX′(B ′ + G)) = Γ (X,∗ωX′(B ′ + G)), completing the proof.

Now we are in a position to prove that Cohen–Macaulay log canonical singularities areDu Bois.

Theorem 3.16. Suppose X is normal and Cohen–Macaulay and � ⊂ X an effective Q-divisorsuch that the pair KX + � is Q-Cartier. If (X,�) is log canonical, then X has Du Bois singu-larities.

Proof. Use Lemma 3.15 setting B = 0 and note that the map of Corollary 3.10 is an isomorphismoutside the singular locus. Furthermore, both sheaves are reflexive (since they are abstractlyisomorphic by Lemma 3.15), completing the proof. �4. The non-normal case

The aim of this section is to show that Cohen–Macaulay semi-log canonical singularities areDu Bois. Let us begin by recalling some of the relevant definitions.

Definition 4.1. A reduced scheme X of finite type over C is said to be seminormal if everyfinite morphism X′ → X of reduced finite type schemes over C that is a bijection on points is anisomorphism.

Remark 4.2. If one is not working over an algebraically closed field of characteristic zero, oneneeds to alter the above definition somewhat. See [8] for details.

Definition 4.3. If X is a reduced scheme of finite type over C with normalization η : XN → X,then the conductor ideal sheaf of X in its normalization is defined to be the ideal sheafAnnOX

(η∗OXN /OX).

Remark 4.4. Consider the affine case where R is a subring of its normalization RN (in its totalfield of fractions). Then the conductor ideal is the largest ideal of RN that is contained in R. This




implies that, with the previous notation, if IC is the conductor ideal sheaf of X in its normaliza-tion and if IB is the extension of IC to the normalization (that is IB = ICOXN ), then η∗IB = IC .

Remark 4.5. If X is seminormal, then the conductor ideal sheaf of X in its normalization is aradical ideal sheaf, even when extended to the normalization, see [33, Lemma 1.3]. If X is S2,then all the associated primes of the conductor IC are height one, see [8, Lemma 7.4], thus all theassociated primes of IB are also height one (cf. [5, 9.2]). Therefore, if X is seminormal and S2,then suppC is exactly the codimension 1 locus of the singular set of X.

Definition 4.6. Let X be a reduced equidimensional scheme of finite type over C. Assume thatX satisfies the following conditions:

(i) X is S2, and(ii) X is seminormal.

These conditions imply that the conductor of X, in its normalization XN , is a reduced idealsheaf corresponding to an effective divisor on XN (cf. (4.5)). We let B denote this divisor onXN and let C denote the corresponding divisor on X (by construction, these divisors have thesame ideal sheaf in the normalization of OX). Further, let � be an effective Q-divisor on X andassume that � and C have no common components. Then (X,�) is said to be weakly semi-logcanonical if the pair (XN,B + η−1∗ �) is log canonical.

Remark 4.7. The log canonical assumption implies that B has to be reduced, but actually thisdoes not impose any new conditions because simply from the fact that X is seminormal, it followsby [8, 1.4] that both B and C are reduced.

Remark 4.8. The usual notion of semi-log canonical also adds the additional condition

(iii-a) X is Gorenstein in codimension 1.

This condition, under the previous seminormality assumption is equivalent to

(iii-b) X has simple double normal crossings in codimension 1.

We won’t need this condition, so we will leave it out. Note that many easy to constructschemes satisfy conditions (i) and (ii) but not (iii). For example, the reduced scheme consist-ing of the three axes in A3 does not have double crossings in codimension 1, but is both S2 andseminormal.

The key ingredient in the proof of the normal case is Proposition 3.4, the injectivity of a certainmap. We will now prove a strengthening of a special case of that injectivity that we will need inthis section. First we need a lemma.

Lemma 4.9. Let d = dimX and Ω×X a complex that completes α to an exact triangle:

OXα

Ω0X Ω×

X

+1.




Then dim Supp(hi(Ω×X)) � d − i − 1 for i � 0 and hi(Ω×

X) = 0 for i < 0.

Proof. The statement follows from the definition for i < 0, so we may assume that i � 0. Fur-thermore, for i = 0 the result also follows since X is reduced and thus X is generically smoothand hence generically Du Bois, so OX → h0(Ω0

X) is an isomorphism outside a set of codimen-sion 1. We proceed by induction on the dimension of X. Clearly it is true for zero (or even one)dimensional varieties (see [4, 4.9] for the one-dimensional case). Let Σ denote the singular setof X and let π : X → X be a resolution of X coming from an embedded resolution as in (3.2)(so that π is an isomorphism over the smooth locus of X). Let E = π−1(Σ)red, that is, E is thereduced pre-image of the singular set. We then have the exact triangle (cf. (3.2)),

Rπ∗OX(−E) Ω0X Ω0

Σ

+1. (4.9.4)

The case i = 0 follows from the fact that h0(Ω0X) is the structure sheaf of the seminormalization

of X by [23, 5.2] or [26, 5.6]. For i > 0, we note that it is sufficient to prove the statement forhi(Ω0

X) ∼= hi(Ω×X). Observe that

dim Supp(

Riπ∗OX(−E))� d − i − 1

because π |X is birational and the dimension of the exceptional set cannot be “too” big (cf. [11,III.11.2]). By the inductive hypothesis the statement holds for hi(Ω0

Σ) and so the long exactsequence coming from the triangle (4.9.4) completes the proof. �

Now we are in position to prove the desired injectivity statement.

Proposition 4.10. In addition to (3.2), let dimX = d . Then the map R−dπ∗ω·X

→ h−d(ω·X) is

injective.

Proof. By local duality, it is enough to show that Hdx (X,OX) → Hd

x(X,Ω0X) is surjective for

every closed point x ∈ X. Considering the exact triangle,

OX Ω0X Ω×

X

+1

shows that it is enough to prove that Hdx(X,Ω×

X) = 0. First observe that Hpx (X,hq(Ω×

X)) = 0for p > dim Supp(hq(Ω×

X)). Furthermore, by Lemma 4.9, we obtain that Hpx (X,hq(Ω×

X)) = 0for p > d − q − 1, and therefore Hd

x(X,Ω×X) = 0 since for p + q = d > d − 1 we see that every

term in the standard spectral sequence that might contribute is already zero. �The next proposition is the key step in our proof that Cohen–Macaulay semi-log canonical

singularities are Du Bois. This can be thought of as a generalization of certain aspects of theKempf-like criterion, Theorem 3.1.

We work in the following setting: X is a reduced d-dimensional Cohen–Macaulay schemeof finite type over C. Let π : X → X be a log resolution of X, and Σ ⊆ X a reduced closedsubscheme such that π is an isomorphism outside Σ (in particular Σ ⊇ SingX). Let F denote




the reduced pre-image of Σ in X. Consider the natural map IΣ → Rπ∗OX(−F) where IΣ isthe ideal sheaf of Σ . Apply R Hom·

X(_,ω·X) = R Hom·

X(_,ωX[d]), which gives us a map

R Hom·X

(Rπ∗OX(−F),ωX[d]) → R Hom·

X

(IΣ,ωX[d]).

Then, by Grothendieck duality,

R Hom·X

(Rπ∗OX(−F),ωX[d]) ∼= Rπ∗R Hom·

X

(OX(−F),ωX[d]) ∼= Rπ∗ωX(F )[d].

Taking the −d th cohomology gives us a natural map

h−d(

Rπ∗ωX(F )[d]) ∼= π∗ωX(F ) → HomX(IΣ,ωX).

We will use properties of this map to deduce that X has DB singularities.

Theorem 4.11. Suppose we are in the setting described above. If the natural map : π∗ωX(F ) →HomX(IΣ,ωX) is an isomorphism, then X has DB singularities.

Remark 4.12. Notice that if X is not normal then Σ contains the conductor.

Proof. By Lemma 3.12 the isomorphism class of π∗ωX(F ) is independent of the choice of theresolution, thus we may assume it came from an embedded resolution of X in some Y as in (3.2).In particular we have π : X → X, where E is the reduced pre-image of Σ in Y (i.e., E|X = F ).First, consider the following map of exact triangles,

IΣ OX OΣ

+1

Rπ∗OX(−F) Rπ∗OX Rπ∗OE

+1.

Applying R Hom·X(_,ω

·X) produces

Rπ∗ω·E

Rπ∗ω·X

Rπ∗ω·X(F )

+1

ω·Σ ω·

X R Hom·X(IΣ,ω·

X)+1

.

Considering the long exact cohomology sequence and using Corollary 3.7 leads to the followingdiagram:

0 h−d(Rπ∗ω·X)

α

h−d(Rπ∗ω·X(F ))

β

h−d+1(Rπ∗ω·E)

γ

. . .

0 h−d(ω·X) h−d(R Hom·

X(IΣ,ω·X)) h−d+1(ω·

Σ) .




Note that β is simply the map and thus it is surjective by hypothesis. Note further that γ

is injective by Proposition 4.10, thus α is surjective by the five lemma. Combining this withProposition 3.4 completes the proof. �

We also need the following two lemmata.

Notation 4.13. Let S be a reduced quasi-projective scheme of finite type of dimension e over C.Then denote by Se the union of the e-dimensional irreducible components of S, and by S<e theunion of the irreducible components of S whose dimension is strictly less than e.

Lemma 4.14. Let Σ be a reduced quasi-projective scheme of finite type of dimension e over C.Then h−e(ω·

Σ) � ωΣe = h−e(ω·Σe

).

Proof. Obviously, dimΣe = e, dimΣ<e < e and dim(Σe ∩ Σ<e) < e − 1. Consider the shortexact sequence

0 → OΣ → OΣe ⊕ OΣ<e → OΣe∩Σ<e → 0

where Σe ∩ Σ<e is not necessarily a reduced scheme. Next, apply R Hom·Σ(_,ω

·Σ) to get a long

exact sequence:

· · · → h−e(ω·

Σe∩Σ<e

) → h−e(ω·

Σe⊕ ω·

Σ<e

) → h−e(ω·

Σ

) → h−e+1(ω·Σe∩Σ<e

) → ·· · .

As dim(Σe ∩ Σ<e) < e − 1, it follows that h−e+1(ω·Σe∩Σ<e

) = h−e(ω·Σe∩Σ<e

) = 0, and hencethe statement holds. �Lemma 4.15. Under the conditions of Theorem 4.11 and using the notation of (4.5) let η : XN →X be the normalization of X and assume that Σe = C where e = dimΣ . Then

η∗ωXN (B) � η∗ HomXN (IB,ωXN ) � HomX(IC,ωX) � HomX(IΣ,ωX).

Proof. The first isomorphism holds because ωXN is a reflexive sheaf and XN is normal. Thesecond follows from Grothendieck duality applied to the finite morphism η (and the definition ofthe conductor). To prove the last isomorphism, consider the map of short exact sequences

0 IΣ OX OΣ 0

0 IC OX OC 0

and apply R Hom·X(_,ω

·X). By Lemma 4.14, we obtain

h−d+1(R Hom·X

(OΣ,ω·

X

)) = h−d+1(ω·Σ

) � h−d+1(ω·C

) = h−d+1(R Hom·X

(OC,ω·

X

)),

thus we have the following diagram:





X(IC,ω·X)) h−d+1(ω·

C)

�

h−d+1(ω·X)


X(IΣ,ω·X)) h−d+1(ω·

Σ) h−d+1(ω·X).

The statement then follows from the five lemma. �Theorem 4.16. If (X,�) is a Cohen–Macaulay weakly semi-log canonical pair, then X has DBsingularities.

Proof. In our setting we may assume that X is non-normal but that it is S2 and seminormal. ThusC, the non-normal locus of X, is a codimension 1 subset of X (see Remark 4.5), in particularC �= 0. Using the same notation as above, let Σ = SingX. Also observe that any log resolutionπ : X → X factors through η:

Xζ

π

XNη

X.

Therefore π∗ωX(F ) = η∗ζ∗ωX(F ). Recall from Remark 4.7 that C is reduced, so Σe = C, andthen by Lemma 4.15, η∗ωXN (B) � HomX(IΣ,ωX). Then by Theorem 4.11 it is sufficient toshow that ζ∗ωX(F ) = ωXN (B). Notice that by definition F = ζ−1∗ B +G where G is the reducedexceptional divisor of ζ (cf. Remark 4.12). By assumption (XN,B + η−1∗ �) is log canonical, soLemma 3.15 implies that ζ∗ωX(F ) = ωXN (B). �5. Cohomologically insignificant degenerations

DB singularities were originally defined by Steenbrink in a Hodge-theoretic context and theyadmit many interesting properties. In particular, they are strongly connected with cohomologi-cally insignificant degenerations.

Inspired by Mumford’s definition of insignificant surface singularities Dolgachev [3] defineda cohomologically insignificant degeneration as follows:

Let f : X → S be a proper holomorphic map from a complex space X to the unit disk S

that is smooth over the punctured disk S \ {0}. For t ∈ S denote Xt = f −1(t). The fiber X0, thespecial fiber, may be considered as a degeneration of any fiber Xt , t �= 0. Let βt : Hj(X ) →Hj(Xt ) be the restriction map of the j th-cohomology spaces with real coefficients. Because X0is a strong deformation retract of X the map β0 is bijective. The composite map

spjt = βt ◦ β−1

0 : Hj(X0) → Hj(Xt )

is called the specialization map and plays an important role in the theory of degenerations ofalgebraic varieties. According to Deligne for every complex algebraic variety Y the cohomologyspace Hn(Y ) admits a canonical and functorial mixed Hodge structure. However in general spj

t isnot a morphism of these mixed Hodge structures. On the other hand, Schmid [24] and Steenbrink[29] introduced a mixed Hodge structure on Hi(X0), the limit Hodge structure, with respect towhich spj

t becomes a morphism of mixed Hodge structures.




In the above setup, X0 is called a cohomologically j -insignificant degeneration if spjt induces

an isomorphism of the (p, q)-components of the mixed Hodge structures with pq = 0. Note thatthis definition is independent of the choice of t �= 0. Finally, X0 is called a cohomologicallyinsignificant degeneration if it is cohomologically j -insignificant for every j .

For us the relevance of this notion is that Steenbrink [30] proved that every proper, flat de-generation f over the unit disk S is cohomologically insignificant provided f −1(0) has DBsingularities. As a combination of Steenbrink’s result and our main theorem, we obtain the fol-lowing.

Theorem 5.1. Let X be a proper algebraic variety over C with Cohen–Macaulay semi-log canon-ical singularities. Then every proper flat degeneration f over the unit disk with f −1(0) = X iscohomologically insignificant.

6. Kodaira vanishing

Following ideas of Kollár [17, §9], we give a proof of Kodaira vanishing for log canonical sin-gularities. This was recently also proven by Fujino using a different technique [7, Corollary 5.11].Our proof applies to (weakly) semi-log canonical singularities as well, while Fujino’s does notuse the Cohen–Macaulay assumption. Partial results were also obtained by Kollár [17, 12.10]and Kovács [21, 2.2].

Convention 6.1. For the rest of the section, all cohomologies are in the Euclidean topology.Nonetheless, the results remain true for coherent cohomology by Serre’s GAGA principle.

First we need a variation on an important theorem.

Theorem 6.2. (See [17, Theorem 9.12].) Let X be a proper variety and L a line bundle on X. LetL n � OX(D), where D = ∑

diDi is an effective divisor (the Di are the irreducible componentsof D). We also assume that the generic point of each Di is a smooth point of X. Let s be aglobal section of L n whose zero divisor is D. Assume that 0 < di < n for every i. Let Z be thescheme obtained by taking the nth root of s (that is, Z = X[√s] using the notation from [17,9.4]). Assume further that

Hj(Z,CZ) → Hj(Z,OZ)

is surjective. Then for any collection of bi � 0 the natural map

Hj(X,L −1

(−

∑biDi

))→ Hj

(X,L −1)

is surjective.

The aforementioned variation of this theorem differs from the version stated above in that Z

was defined to be the normalization of X[√s ]. However, as we are dealing with possibly non-normal schemes (e.g., slc) we need this version. In some sense, the proof of this formulationis actually easier and we sketch the argument below. The strategy is the same as in [17, Theo-rem 9.12], but as Z is not normalized, some steps and ingredients are different.




Proof. Let Z = X[√s ] = SpecX

∑n−1t=0 L −t and p : Z → X denote the natural map. By con-

struction there is a decomposition p∗OZ � ∑n−1t=0 L −t . Fixing an nth root of unity ζ and

considering the associated Z/n-action on Z (coming from the cyclic cover), we see that Z/n actson the summand L −t by multiplication by ζ−t , so p∗OZ � ∑

L −t is actually the eigensheafdecomposition of p∗OZ , cf. [17, Proposition 9.8]. One also has an eigensheaf decomposition,p∗CZ � ∑n−1

t=0 Gt . For ease of reference, set Gt to be the eigensheaf corresponding to the eigen-value ζ−t . In particular, Gt ⊆ L −t .

With these decompositions, note that we have a surjective map

∑t

H j (X,Gt ) � Hj(Z,CZ) � Hj(Z,OZ) �∑

t

H j(X,L −t

)

and so in particular we have a surjection

Hj(X,G1) � Hj(X,L −1) (6.2.1)

for every j . We now claim that G1 is a subsheaf of L −1(−∑biDi). As they are both sub-

sheaves of L −1, this is a local question and it is enough to show that for every connectedopen set U ⊆ X, the inclusion γ : Γ (U,G1) ↪→ Γ (U,L −1) factors through the inclusionδ : Γ (U,L −1(−∑

biDi)) ↪→ Γ (U,L −1).If U is such that U ∩ D = ∅, then δ is an isomorphism and so the statement holds trivially.If U is such that U ∩ D �= ∅, then Γ (U,G1) = 0 by (6.2.2), and so the statement again holds

trivially.

Claim 6.2.2. Let U ⊆ X be a connected open set such that U ∩ D �= ∅. Then Γ (U,G1) = 0.

Proof. We give two short proofs of this claim.First, let Z denote the normalization of Z, p : Z → X the induced map, and Gt the eigensheaf

of the Z/n action on p∗CZ corresponding to the eigenvalue ζ−t . Then p∗CZ ↪→ p∗CZ naturally,so in particular, Gt ⊆ Gt for all t and hence the statement follows by [17, 9.11.3].

Alternatively, one can give a direct proof as follows. The assumptions imply that there existsa dense open subset U ′ ⊆ U such that each x ∈ U ′ ∩ D has a neighborhood where X is smoothand D is defined by a power of a coordinate function. Then the computation in [17, 9.9] showsthat p−1U ′ and therefore p−1U are connected and the claim follows easily. �

Therefore G1 is indeed a subsheaf of L −1(−∑biDi) and one obtains a factorization

Hj(X,G1) Hj (X,L −1(−∑biDi)) Hj (X,L −1),

where the composition is surjective by (6.2.1) and so the second arrow is surjective as well. �Theorem 6.3 (Serre’s vanishing). (See [19, 5.72].) Let X be a projective scheme over a field k

of pure dimension n with ample Cartier divisor D. Then the following are equivalent:

(1) X is Cohen–Macaulay,(2) Hj(X,OX(−rD)) = 0 for every j < n and r � 0.




In order to use the “usual” covering trick, we need to establish that our assumptions are in-herited by the covers. Examples of rational singularities with non-Cohen–Macaulay canonicalcovers in [28] suggest that this is not entirely obvious.

First we need the following construction.

Notation 6.4. Let τ : S → T be a finite morphism between reduced schemes of finite type overC and assume that T and S are normal. Let i : U = T \ SingT ↪→ T and j : V = τ−1U ↪→ S.Since T is normal and τ is finite, codimT (T \ U) � 2 and codimS(S \ V ) � 2. Let D ⊆ T be aneffective Weil divisor. Then D|U is a Cartier divisor corresponding to the invertible sheaf L onU and D induces a section of L : δ : OU ↪→ L . Furthermore, OT (D) � i∗L . Then the pullbackof δ induces a section τ ∗δ : OV ↪→ τ ∗L , which in turn induces a section of the rank 1 reflexivesheaf j∗τ ∗L on S. Denote the corresponding Weil divisor by τ [∗]D, i.e., OS(τ [∗]D) � j∗τ ∗L .An alternative way to obtain τ [∗]D is to take the closure (in S) of the divisor τ ∗(D|U) ⊆ V .

Lemma 6.5. Let (X,�) be a projective log variety with weakly semi-log canonical singularitiesand σ : Z → X a cyclic cover of X induced by a general section of a sufficiently large power ofan ample line bundle as in [19, 2.50]. Then there exists an effective Q-divisor Γ on Z such that(Z,Γ ) is weakly semi-log canonical. Furthermore, if X is Cohen–Macaulay, then so is Z.

Proof. Let L be an ample line bundle on X, m � 0 and s ∈ H 0(X,L m) a general section.Then D = (s = 0) is reduced. As before, let η : XN → X denote the normalization of X andB ⊂ XN the extension of the conductor to XN (cf. (4.5)). Then, by assumption, (XN,B +η−1∗ �)

is log canonical. Observe that as η is finite, η∗L is also ample. Therefore, by [19, 5.17.(2)],(XN,B + η−1∗ � + η∗D) is also log canonical.

Let A = ⊕m−1i=0 L −i with the OX-algebra structure induced by s. Let σ : Z = SpecX A → X

be as in [19, 2.50] and similarly σ : W = SpecX η∗A → XN .

Wσ

τ

XN

η

Zσ

X.

By assumption XN is normal, i.e., R1 and S2 by Serre’s criterion. Then W is also R1 by[19, 2.51] and furthermore σ∗OW � η∗A is also S2 (as it is locally free), so we see that W isalso S2 by [19, 5.4]. Therefore W is normal by Serre’s criterion and hence τ factors through thenormalization of Z:

Wτ

τ

Z Z.

From the construction it is clear that τ : W → Z is birational, and hence so is τ . However, thenit must be an isomorphism by Zariski’s Main Theorem, and hence W is the normalization of Z.

As X is S2, so is σ∗OZ � A , and then Z is S2 as well.Let T = (SingX ∩ D)red ⊆ X and T = σ−1T ⊆ Z closed subsets in X and Z respectively.




Then by construction codimX T � 2 and hence codimZ T � 2. Observe that for any z ∈ Z \ T

either Z is smooth at z or σ is étale in a neighborhood of z in Z. Therefore, Z is seminormal incodimension 1 (since X is seminormal) and as we have just shown that Z is also S2, it followsby [8, 2.7] that Z is seminormal everywhere.

Let M = (σ ∗(η∗D))red. Since XN and W are smooth in codimension 1, and σ is ramifiedexactly along η∗D with degree m and ramification index m everywhere, it follows, that

σ ∗(KXN + B + η−1∗ � + η∗D) = KW + σ [∗](B + η−1∗ �

) + M.

Then (W, σ [∗](B + η−1∗ �) + M) is log canonical by [19, 5.20(4)].Let BZ denote the subscheme defined by the conductor of Z in W . We claim that BZ is

contained in the proper transform of B , i.e., BZ ⊆ σ [∗]B (cf. (4.4)). To see this, note that since Z

is S2 and seminormal, the conductor is simply the codimension 1 part of the non-smooth locusof Z. But the non-smooth locus of Z is the pre-image of the non-smooth locus of X by [19, 2.51]and so the claim follows.

Then there exists an effective Q-divisor Θ on W satisfying σ [∗](B +η−1∗ �) = BZ +Θ . Now,if we choose Γ = τ∗(Θ + M), then (Z,Γ ) has weakly semi-log canonical singularities.

If X is Cohen–Macaulay, then so is A � π∗OZ and Z is Cohen–Macaulay by [19, 5.4]. �Corollary 6.6. Kodaira vanishing holds for Cohen–Macaulay weakly semi-log canonical vari-eties: Let (X,�) be a projective Cohen–Macaulay weakly semi-log canonical pair and L anample line bundle on X. Then Hi(X,L −1) = 0 for i < dimX.

Proof. We will use the notation from (6.5). Z is Cohen–Macaulay and (Z,Γ ) is weakly semi-log canonical. Therefore by (4.16) Z is Du Bois, and it follows from (6.2) that Hi(X,L −m) →Hi(X,L −1) is surjective for all m � 0. Serre’s vanishing (6.3) implies that Hi(X,L −m) = 0for m � 0 and i < dimX, so the desired statement follows. �

This implies invariance of plurigenera in stable Gorenstein families.

Corollary 6.7. Let f : X → S be a stable Gorenstein family, i.e., a flat projective family ofcanonically polarized varieties with at most Gorenstein (weakly) semi-log canonical singulari-ties. Then hi(Xt ,ω

mXt

) is independent of t ∈ S for any m > 0 and i � 0.

Proof. By Serre duality hi(Xt ,ωmXt

) = hdimXt−i (Xt ,ω−m+1Xt

) and the latter vanishes for i > 0and m > 1 by (6.6). Then

h0(Xt ,ωmXt

) = χ(Xt ,ω

mXt

)for m > 1 and this is independent of t because f is flat.

So we may assume that m = 1. Then hi(Xt ,ωXt) = hdimXt−i (Xt ,OXt

) and this is inde-pendent of t for any i by (4.16) and [4, Théorème 4.6]. �References

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